Question

Consider the first five steps of the derivation of the Quadratic Formula.

What is the next step?

(See the photo)

Answer 1

Answer:

B

Step-by-step explanation:

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Answer 2

The method of completing the squares method will be used for the next step. The next step would be ( x+ (b/2a)  )^2 = -c/a + b^2/(4a^2)

What is the expansion of square of sum of two terms?

Suppose that two terms are 'a' and 'b'.

Then, their sum's square is expanded as:

$(a+b)^2 = a^2 + 2ab + b^2$

For this case, we're given the quadratic equation with its 4 steps of solution as:

$ax^2 + bx + c = 0\\\\\dfrac{ax^2}{a} + \dfrac{bx}{a} + \dfrac{c}{a} = 0\\\\x^2 + \dfrac{bx}{a} + \dfrac{c}{a} = 0\\\\x^2 + \dfrac{bx}{a} = -\dfrac{c}{a}\\\\x^2 + \dfrac{bx}{a} + \dfrac{b^2}{4a^2} = -\dfrac{c}{a} + \dfrac{b^2}{4a^2}$

We've to proceed to the next step.

In the left hand side, we can use the formula $(a+b)^2 = a^2 + 2ab + b^2$, where $\rm a \: is \: x^2, and \: b \: is \: \dfrac{b}{2a}$

Thus, we have:

$x^2 + \dfrac{bx}{a} + \dfrac{b^2}{4a^2} = x^2 + 2 \times x \times \dfrac{b}{2a} + \left(\dfrac{b}{2a} \right)^2 = \left( x + \dfrac{b}{2a} \right)^2$

We expressed the left sided expression in this form so that the variable 'x' (which is assuminly the only variable in the equation, whose value is to be known) gets packed into single container and not spread throughout with different powers.

Now, we can use square root and then subtraction and division of some terms which will make 'x' stay on one side of the equation, and rest of the constants on other side, as the value of 'x' for which the considered equation $ax^2 + bx + c = 0$ is correct.

The next step of the solution, therefore, is:

$\left( x + \dfrac{b}{2a} \right)^2 = \dfrac{-c}{a} + \dfrac{b^2}{4a^2}$

Learn more about completing the squares method here:

https://brainly.com/question/16800259

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